18.S34 (FALL, 2007)
PIGEONHOLE PROBLEMS
Note: Notation such as (78P) means a problem from the 1978 Putnam
Exam.
1. (78P) Let A be any set of 20 distinct integers chosen from the arith­
metic progression 1, 4, 7, . . . , 100. Prove that there must be two distinct
integers in A whose sum is 104. [Actually, 20 can be replaced by 19.]
2. Five points are situated inside an equilateral triangle whose side has
length one unit. Show that two of them may be chosen which are less
than one half unit apart. What if the equilateral triangle is replaced
by a square whose side has length of one unit?
3. (71P) Let there be given nine lattice points (points with integral co­
ordinates) in three dimensional Euclidean space. Show that there is a
lattice point on the interior of one of the line segments joining two of
these points.) [To test your understanding, how many lattice points
does one need in four dimensions to reach the same conclusion?]
4. (72IMO) Prove that from a set of ten distinct two-digit numbers (in
the decimal system), it is possible to select two disjoint subsets whose
members have the same sum. [Though not stated in the problem, one
should assume that not both the subsets are empty, or even that neither
of the subsets is empty.]
5. (80P) Let A1 , A2 , . . . , A1066 be subsets of a finite set X such that |Ai | >
1
|X| for 1 � i � 1066. Prove that there exist ten elements x1 , . . . , x10
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of X such that every Ai contains at least one of x1 , . . . , x10 . (Here |S |
means the number of elements in the set S.)
6. Given any n + 2 integers, show that there exist two of them whose sum,
or else whose difference, is divisible by 2n.
7. Given any n + 1 distinct integers between 1 and 2n, show that two
of them are relatively prime. Is this result best possible, i.e., is the
conclusion still true for n integers between 1 and 2n?
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8. Given any n + 1 integers between 1 and 2n, show that one of them is
divisible by another. Is this best possible, i.e., is the conclusion still
true for n integers between 1 and 2n?
9. Given any 2n − 1 integers, show that there are n of them whose sum
is divisible by n. (Though superficially similar to some other pigeon­
hole problems, this problem is much more difficult and does not really
involve the pigeonhole principle.)
10. Is it possible to cut an 8 × 8 chessboard with 13 straight lines (none
passing through the midpoint of a square) such that every piece con­
tains at most one midpoint of a square?
11. Let each of nine lines cut a square into two quadrilaterals whose areas
are in the proportion 2 : 3. Prove that at least three of the lines pass
through the same point.
12. Let a1 < · · · < an , b1 > · · · > bn , and {a1 , . . . , an , b1 , . . . , bn } =
{1, 2, . . . , 2n}. Show that
n
�
i=1
|ai − bi | = n2 .
13. Let u be an irrational real number. Let S be the set of all real numbers
of the form a + bu, where a and b are integers. Show that S is dense in
the real numbers, i.e., for any real number x and any � > 0, there is a
element y � S such that |x − y| < �. (Hint. First let x = 0.)
14. Two disks, one smaller than the other, are each divided into 200 congru­
ent sectors. In the larger disk 100 of the sectors are chosen arbitrarily
and painted red; the other 100 sectors are painted blue. In the smaller
disk each sector is painted either red or blue with no stipulation on
the number of red and blue sectors. The small disk is then placed on
the larger disk so that their centers coincide. Show that it is possible
to align the two disks so that the number of sectors of the small disk
whose color matches the corresponding sector of the large disk is at
least 100.
15. A collection of subsets of {1, 2, . . . , n} has the property that each pair
of subsets has at least one element in common. Prove that there are at
most 2n−1 subsets in the collection.
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16. (95P) For a partition � of {1, 2, 3, 4, 5, 6, 7, 8, 9}, let �(x) be the number
of elements in the part containing x. Prove that for any two partitions �
and � � , there are two distinct numbers x and y in {1, 2, 3, 4, 5, 6, 7, 8, 9}
such that �(x) = �(y) and � � (x) = � � (y).
[A partition of a set S is a collection of (nonempty) disjoint subsets
(parts) whose union is S.]
17. (a) Let p be a prime of the form 8k +1. Prove that there exist positive
integers a, b, m with m < 2p such that mp = a4 + b4 .
Hint. Show that there is an integer x such that x4 + 1 is divisible
�
by p, and consider the numbers u + vx, where 0 � u � ≤ p√ and
�
0 � v � ≤ p√.
Note. This is a minor variation of a standard application of the
pigeonhole principle going back to Fermat. Do not hand in this
problem if you’ve seen it or something similar before.
(b) Improve the bound m < 2p. In particular, find a constant c < 2
such that one can take m < cp for p large. (The best possible value
of c requires some sophisticated number theory not involving the
pigeonhole principle.)
18. N is the set of nonnegative integers. For any subset S of N, let P (S)
be the set of all pairs of members of S. (A pair is a set (unordered)
with two distinct members.) Partition P (N), arbitrarily, into two sets
(of pairs) P1 and P2 . Prove that N must contain an infinite subset S
such that either P (S) is contained in P1 or P (S) is contained in P2 .
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